NASA AVSCOMTechnical Memorandum 103798 Technical Report 91-C-008

Computerized Inspection of RealSurfaces and Minimization ofTheir Deviations

AD-A240 066

F.L. Litvin, Y. Zhang, and C. KuanUniversity of Illinois at Chicago 0Chicago, Illinois V1 F. KCT r

and . P.O 4 199!,

R.F. HandschuhPropulsion DirectorateU.S. Army Aviation Systems Corn mandNASA Lewis Research CenterCleveland,. Ohio

Prepared for the5th International Conference on Metrology and Propertiesof Engineering SurfacesLeicester Polytechnic, England, April 10-12, 1991

91-09449 US ARMY

Il SYSTEMS COMMAND

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QUALITY AVAILABLE. THE COPY

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COMPUTERIZED INSPECTION OF REAL SURFACES AND MINIMIZATION OF THEIR DEVIATIONS

F.L. Litvin, Y. Zhang, and C. Kuan

University of Illinois at Chicago

Chicago, Illinois 60680

and

R.F. Handschuh

Propulsion DirectorateU.S. Army Aviation System Command

Lewis Research Center

Clevelaii, Ohio 44135

SUMMARY

A method is developed for the minimization of gear tooth surface deviations

between theoretical and real surfaces to improve the precision of surface manufac-ture. Coordinate measurement machinery is used to determine a grid of surfacecoordinates. Theoretical calculations are made for the grid points. A least-squaremethod is used to minimize the deviations between real and theoretical surfaces byaltering the manufacturing machine-tool settings. An example is given for a hypoidgear.

INTRODUCTION

The Gleason Works were pioneers in the application of coordinate measurements toimprove the precision manufacturing of hypoid and spiral bevel gears (ref. 1). Inaerospace applications, duplication of flight-qualified master gears is very impor-tant, and coordinate measurement has now become part of the normal production pro-cess. Methods to enhance and extend the use of this machinery can be very valuableto aerospace gear manufacturers.

The approach developed in this paper enables one do determine deviations of areal surface from the known theoretical surface. This is accomplished by usingcoordinate measurements and minimizing the deviations to correct the previously

applied machine-tool settings. The surface deviations are represented in the direc-tion of the normal to the theoretical surface. The coordinate measurements areperformed by a machine with 4 or 5 degrees of freedom. In the case of 4 degrees offreedom, the probe performs three translational motions (fig. 1); the fourth motion,rotation, is performed by turning the table with the workpiece. The axis ofrotational motion coincides with the axis of the workpiece. In the case of a5-degree-of-freedom machine, the fifth degree of freedom is used to provide the probedeflections in the direction of the normal to the theoretical surface. The probe isprovided with a changeable spherical surface whose diameter can be chosen from a widerange.

The motions of the probe and the workpiece for coordinate measurements are com-puter controlled and for this purpose a grid, the set of points on the surface to bemeasured, must be chosen. There is a reference point, one point on the grid, that isnecessary for the initial installments of the probe. There are two orientations ofthe probe installment that are applied for measurements of a gear (fig. l(a)) and apinion (fig. 1(b)), depending on the anglh Z ieit cone.

- ProbeZm'I

Gear--

. ..

Backface- /base plane- /

"- Rotary table(a)

/-Probe

Pinion .

Backfaf- surac ca be d e t n hffmbae p he tool s N t ru

__Rotary table(b)

Figure 1-Surface measurement orientation for (a) gearand (b) pirion.

The mathematical aspects of coordinate measurements will now be described

(ref. 2): First, it is necessary to derive the equations for the theoretical sur-

face. In many cases, this surface can be derived as the envelope to the family of

generating surfaces, namely the tool surfaces. Next, the results of coordinatemeasurements must be transformed into deviations of the real surface represented in

the direction of the surface normal. Here, the surface variations are represented in

terms of the corrections to the machine-tool settings. The surface deviations

obtained from coordinate measurements and the surface variations determined by the

corrections of machine-tool settings can be represented by an overdetermined system

of linear equations. The number k of these equations is equal to the number ofgrid points, and the number of unknowns m is equal to the number of corrections of

machine-tool settings (m << k). The optimal solution to such a system of linearequations enables one to determine the sought-for corrections of machine-tool

settings.

EQUATIONS OF THEORETICAL TOOTH SURFACE E.t

Considering that the theoretical surface can be determined directly, we repre-

sent it in coordinate system S in two-parametric form ast

where r and n are the position vector and unit normal to the surface, respec-t )t

tively, and (u,G) are the Gaussian coordinates (surface coordinates).

2

For the case when surface is the envelope to the family of generating- t

surface Lc, we represent in S surface E and the unit normal n to E as(ref. 3)

r t = [Stjrc(Uc, ), = 0 (2)

t = [Ltc]n,(u. ,c),f(uc, , ) = 0 (3)

where (uc,c) are the Gaussian coordinates of the generating surface E , and 0 isthe generalized parameter of motion in the process fo" --reration. The equation ofmeshing is

4 () ct) 4-1

where N(C) is the normal to E', and v (ct) is the relative motion for a point ofcontact of £E and E . Matrices [M tc] and [Lt ] 3x3 describe the coordinatetransformation from S to S for a position vector and surface normal, respec-C t

tively. Position vectors in three-dimensional space are represented with homogeneous

coordinates.

COORDINATE SYSTEMS USED FOR COORDINATE MEASUREMENTS

Coordinate systems Sm and St are rigidly connected to the coordinate measur-ing machine (CMM) and the workpiece being measured, respectively (fig. 2). The

Xm

• i O t

Aoession For

NTIS GRA&IDTIC TAB 13

Figure 2 -Relationship between theoretical and measurement Unannounced 0coordinate systems. (p, radial distance to point from axis ofrotation.) Just if I cat ton -

By --------- _ ___

Distribution/

I Availability Codes

Avail and/or

Dis Special

3

Z t

O tl iii Yt

Fig,,re 3.-Measurement grid on tooth surface.

backface cf the gear is installed flush with the base plane of CMM. The distance 1between the origins 0 and 0 is assumed to be known, but the parameter 6 ofm t

orientation must be determined (see the following section). The coordinate transfor-mation from S to S is represented by the matrix equation

r m = [Mt]r t (5)

MEASUREMENT GRID AND ESTABLISHMENT OF THE REFERENCE POINT

The grid is a set of points on E chosen as points of contact between theprobe and E. (fig. 3). Fixing the value of zt for the point of the grid and thevalue of, say, Y. (or x t), we can obtain the following equations:

y (ui,e ) = h, z(ui,oi) = li (i = 1 .... k) (6)

where k is the number of grid points.

We consider h. and i as given and solve equations (6) for (u.,6). Then wecan determine the position vectors and the unit normals for k points of the gridusing the equations

i) (7)r. - [xt(ui, 1)Yt(Ui,0i)zt(ui ) T, (i = 1 .... ,k)

n (i) [nx(U )n ((u, ) ] T i = 1 ... , k) (8)nt t Ytu'jntu,

The position vector for the center of the probe, if the deviations are zero, isrepresented by

4

R t = rt + pnt (i = 1,. .. k)(9

where p is the radius of the probe sphere.

The referenri point

(0) (0) ) ) (o) T (10r t =[x t u 0(0) )Yt(Uo 0,(o )zt( H( )

is usually chosen as the mean point of the grid.

The center of the probe that corresponds to the reference point on Et isdetermined from equation (9) as

(0)) u(0) u(0)) T(iRt = [Xt(u ( ,0 (0 )Yt(u ,' °) Zt( ' 0)H( 1

where (u are known values.

The coordinates of the reference center of the probe are represented in coordi-nate system S of the measuring machine by the matrix equation

(0) (0) (12)

Equation (12) yields

(0) (o) 0) J

xm = xm (5,u(0, )

(0) ( 0) (13)Ym = ym (6,(u (),

(0) (o) ( (0)z M = z m ( 6 ,u 0

(0) (0) (0)Three equations (13) contain four unknowns: 6, xm ' Y , z. To solve these equa-tions, we may consider that one of the coordinates of the reference point of the

(0)probe center, say, ym, may be chosen equal to zero. This is accomplished byrequiring the reference point to lie in the xm - zm plane. The orientation ofangle 6 is now established to satisfy this requirement, and all measurements arereferenced from this location. Then equation system (13) allows one to determine

(o) (0) (0) (0) (0)6 ,x m and zm (ref. 2). Coordinates xM y = 0, zM are necessary for theinitial installment of the center of the probe.

5

MEASUREMENT OF THE DEVIATIONS OF THE REAL SURFACE

The deviations of the real surface are caused by manufacturing errors, heattreatment, etc. Vector positions of the center of the probe for the theoreticalsurface and the reai surface can be represented a. follows:

Rm = rm(u,6) + pnm(u,) (14)

R* = rT(u,O) + Xn (u,e) (15)

where r and n are the position vector and the unit normal to the theoreticalm m

surface and are represented in coordinate system S of the measuring machine; Xdetermines the real location of the probe center and is considered along the normal

to the theoretical surface; R and R represent in S the position vector of thein mprobe center for the theoretical and real surfaces, respectively. Equations (14)

and (15) yield

R Rm = (X - P)nm = Annm (16)

and

An = (R* - R n m (17)

m M,

The position vector Rm is determined by coordinate measurements for points of thegrid. Equation (17) determines numerically the function

An, = Ani(ui, i) (i = 1 ..... k) (18)

that represents the deviations of the real surface for each point of the grid.

MACHINE TOOL SETTINGS TO MINIMIZE DEVIATIONS

The procedure used to minimize the deviations can be represented in two stages:(1) determination of variations of theoretical surface caused by changes of appliedmachine-tool settings, and (2) minimization of deviations of real surface by appro-priate correction of machine-tool settings.

We consider that the theoretical surface is represented in S ast

rt = rt(uO,dj) (j = 1,... m) (19)

where parameters d are the machine-tool settings. The surface variation is

represented by

art art m art6rt 5u + 60 + t6d (20)

6I

6

We multiply both sides of equation (20) by the surface unit normal nt and take into

account that Ort/3G n = ar./au n = 0, since art/aO and r t/au lie in theplane that is tangent to the surface. Then we obtain

art d a(d (21)

5r t • nt = E nj1d

We can now consider a system of k linear equations in m unknowns (m << k) of thefollowing structure:

al 6d1 + a12 5d 2 + + alin6dn = b1

............................... (22 )

a, 1 5d1 + ak 6 d2 + ++ a6d, =bk

Here

b. = An, = (R.- Ri) ., (23)

where i designates the number of grid points; a (s = 1...,k; j =

represents the dot product of partial derivatives r t/ad j and unit normal n . Thesystem (22) of linear equations is overdetermined since m << k. The essence of theprocedure for miaimization of deviations is determining unknowns 6d (j =that will minimize the difference between the left and right sides of equations (22).The solution employed the least-square method. The subroutine DLSQRR of IMSL MATH/LIBRARY (ref. 4) was used to computerize the procedure.

APPLICATION OF METHOD TO THE INSPECTION OF FORMATE HYPOID GEAR

Each tooth side of a formate face-hobbed gear is generated by a cone, and thegear tooth surface is the surface of the generating cone. Two cones that are shownin figure 4(a) represent both sides of the gear space. The following equationsrepresent in coordinate system S gear surfaces for both sides and the unit normalto such surfaces (fig. 4(b)):

-s~cos aG

(r - sGsin a,)sin 0G (24)r =

(r - sGsin a')cOs 0G

1

• • • 7

F- GeneratingII cones

Cutteru blade - P

H2 Oc ZCr rc

cc t -

GG

xox

(a) (b)Y0 t

Figure 4 -Generating cones representing cutter blades. Figure 5-Coordinate system orientation and machine-toolsettings for hypoid gear.

.08-

04

E -04 Drve side

C0

S 08_0

E 040

-04 Coast side

-.08 1 1 1 1 1 1 1 I _ _0 5 10 15 20 25 30 35 40 45

Number of points

Figure 6 -Deviations of gear real tooth surface.

8

sinl a-,

- cos a sin G _ (25)

- cos acos 6

where, r is the position vector and n the unit normal; r is the cutter tip

radius; is the cutter blade angle (aG > 0 for the concave side and a < 0 for

the convex side).

Figure 5 shows the installment of the generating cone on the cutting machine.

Coordinate systems S and S are rigidly connected to the cutting machine arid the

gear being generated, respectively. Systems S , S , and S are rigidly connectedC

to each 3ther since the gear is formate cut. To represent in St the theoreticalgear tooth surface and the unit normal to E_, w .se the following matrix

equations:

r,(s0 ,eG,d,) = [M,.]r,(sC,OC) (26)

nt(s,&6,d,) = [L,,)n,(s,,6 ) (27)

where

cos'y 0 -sin'7m 0 1 0 0 0

0 1 0 0 0 1 0 -V7 (28)

sin7, 0 cos7m -AXm 0 0 1 H,

0 0 0 1 0 0 0 1

The surface Gaussian coordinates are s and 0., and d.(T , V2, H 2, and AX,) arethe machine-tool settings.

The numerical example presented in this paper is based on the experiment that

has been performed at the Dana Corporation (Fort Wayne, IN, U.S.A.). The initialdeviations An for each side of the real tooth surface have been obtained by

measurements on a coordinate measuring machine (fig. 6). The qrid for the meas-urement is formed by nine sections along the tooth length, each section having five

points. The number of grid points k is therefore 45, and the reference point is atthe middle of the grid, i.e., the third point of the fifth section. In the measure-

(0)

ment, the coordinate y of the reference point is chosen to be zero and thealignment angle 6 is determined from solving equation system (13).

The minimization of deviations was performed in accordance with the algorithm

described in MACHINE TOOL SETINGS TO MINIMIZE DEVIATIONS, and the results areillustrated in figure 7 and table 1.

9

SDrive side Coast sideC0 04 -> E

0 '

0808- I I I I0 to 20 30 40 50 60 70 80 90

Number of points

Figure 7 Minimized deviations after corrections made to machine-tool settings

TABLE I. - RESULTS OF MINIMIZATION

fPressure angle, Ct = 21.250; cutter diameters = 9 in.,

point width of cutters = 0.08 in.]

Machine-tool Machine-tool setting parameters

settings H, Ax

mm mm rad mm

Initial 103.252550 27.466600 1.059816 0.009677

Corrected 103.25220 27.21603 1.06437 -0.53343

CONCLUSION

A general approa-ch for a computerized determination of deviations of a real sur-face from the theoretical one based on coordinate measurements has been proposed. An

algorithm for computerized minimization of deviations by corrections of initiallyapplied machine-tool settings has been developed. The approach is illustrated with

the example of the tooth surface of a hypoid formate gear.

ACKNOWLEDGMENT

This research has received financial support from the NASA Lewis Research

Center, Gleason Memorial Fund, and the Dana Corporation.

REFERENCES

1. Gleason Works: "G-Age T.M. User's Manual," for the Gleason Automated GearEvaluation System Used with Zeiss Coordinate Measuring Machines, 1987.

2. Litvin, F.L.; Zhang, Y.; Kieffer, J.; and Handschuh, R.F.: Identification and

Minimization of Deviations of Real Gear Tooth Surfaces. To be published, J.

Mech. Design, vol. 113, no. 1, Mar. 1991.

3. Litvin, F.L.: Theory of Gearing. NASA RP-1212 (AVSCOM technical report;

88-C-035), 1989.

4. Dongarra, J.J., et aL. : LINPACK User's Guide. SIAM, Philadelphia, 1979.

10

Nbri/lAatsan Report Documentation PageSpaoe Ad-,',ntraton1 Report No NASA TM - 103798 2.Gvrnment Accession No. 3. Recipient's Catalog No.

AVSCOM TR91_- C -008 ___________

4. Title and Subtitle 5. Report Date

ComliuteriZe'd In spectin ito Real Stirfaces and Mlinindz~ationof Thoir Devitions

6. Perforiniuig Organization Code

7 Authorls) 8. Performing Organization Report No

F.L. Larvin,. Y. Zhano, C. Kilian, and R .F. HanldSCltuh -60

tO Work Unit No.

9 Performing Organization Name and Address 505-6 )oN~kSA Lew., R , .earch Ccni f1L16PXA47ACle vland, Ohio 44 1.'5 - 1191and t11. Contract or Grant No.

Prouti tiin DirectoirateU .S. Artov Aviation Systems CommandClevelatnd, Ohio 44135 - 3 19 1 13. Type of Report an'd Perod Cover-A

2 Siorisiirng Agency Naine and Address Technical MemorandumNational Aerionautics ;and Space Admllin istrat ionWashtttgtotn, D.C. 20546 - 0001 14. Sponsoring Agency Code

a ttdU.S. Army A . ion Systems CommandSt, Lolis, Mlo 63 120 - 1798

15 Skippernentari Notes

Prepared [i r the 5th International Conference on Metrology and Properties of Engineering Surfaces, Leicester Polytechnic,E ladApril1 10- 12. 199 1. F.L. Litvin, Y. Zhang, C. Kuan, Dept. of Mechanical Engineering, University of Ilinois at

'llica,:o. Chicago, Illinois 60680 (work funded under NASA Grant NAG3-964). R.F. Handschuh, Propulsion Director-ale, U.S. Artny Aviation Systems Command. Responsible person, R.F. Handschuh, (216) 433-3969.

16 Abstract

A nmethod IS developed for the minimization of gear tooth surface deviations between theoretical and real surfaces forthe tmprovenment of precision of surface - !anufacture. Coordinate measurement machinery is used to determine a gridlif surface coourdinates. Theoretical calculations are made for the grid points. A least-square method is used to minimizethe dev iations between real and theoretical surfaces by altering the manufacturing machine-tool settings. An example isgiven ftor a hvpoid gear.

17 Key Words (Suggested by Author(s)) 18. Distribution Statement

Gears Unclassified - UnlimitedGear I, eth Subject Category 37Mechanical drivesCoo~rdinate transformations

19 Secunity Clasill (of the report) 20. Security Ctassil. (ot this page) 21. No. of pages 22 Price*

Unclassified Unclassified 112 A03

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